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Geodesic Cone in Nil-Geometry


	W. Heisenberg's real matrix group provides a noncommutative translation group of an affine three-space. The Nil-geometry, which is one of the eight Thurston three-geometries, can be derived from this group. E. Molnár proved that the homogeneous three-spaces have a unified interpretation in the projective three-sphere ). In this Demonstration a geodesic line rotated around the axis, (a "geodesic cone") is visualized.


	Contributed by: Benedek Schultz and János Pallagi Suggested by: Jenő Szirmai
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You get a "geodesic cone" by rotating a geodesic curve around the

axis. The geodesic curves of the Nil-geometry are generally defined as having locally minimal arc length between any two (near enough) points. The system of equations of a parametrized geodesic curve is (where

, and

Here

and

are the parameters of a geodesic curve (

); in this Demonstration you can adjust these values.

The following is an example of a geodesic curve with parameters

A point

rotated through

has the following coordinates: (

As you can see, a geodesic curve returns periodically to the

axis. We get the "geodesic cone" by rotating the part of the geodesic curve between the origin and the first return to the

axis around the

axis.

If we rotated the whole curve, then it would look like this (with

In this Demonstration you can adjust the

and

parameters, and according to this, the

parameter (the arc length parameter) has a value

Reference:

J. Szirmai, "The Densest Geodesic Ball Packing by a Type of Nil Lattices," Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry), 48(2), 2007 pp. 383–397.

Geodesic (Wolfram MathWorld)

Geodesic Balls in the Nil-Space (The Wolfram Demonstrations Project)

Nil Geometry (Wolfram MathWorld)

Torus in Nil-Space (The Wolfram Demonstrations Project)

"Geodesic Cone in Nil-Geometry" from The Wolfram Demonstrations Project
http://demonstrations.wolfram.com/GeodesicConeInNilGeometry/

Contributed by: Benedek Schultz and János Pallagi

Suggested by: Jenő Szirmai

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